Announcement: new book being written (by me!)


Roger Bissell

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Folks, I've been a little overly sensitive about being misunderstood, and I ask people's patience and understanding in regard to remarks I've previously made. I benefit from everyone's most intelligent, perceptive comments, as well as suggestions for further listening and analysis. By asking the right questions, or questioning what seems like a wrong-headed idea, at the very least you will alert me to something that I may need to work harder to clarify. And you very well may save me a lot of time in avoiding a blind alley of some kind. That said, I have no reason to think that my approach is basically wrong-headed, just not very well expressed in some respects.

Michael is right in saying that I am definitely in exploratory mode here. I am offering my ideas not as an attempt to prescribe how to think about music, but as an indication of the direction I'm going, and as an opportunity for you readers to chime in with suggestions. Even when I'm convinced about something, I don't always state it clearly and accurately, since there is so much to the subject, and my comments aren't systematically laid out, more off the top of my head at this point.

For instance, the whole discussion of the Rachmaninoff prelude in C#minor's middle section was about the sense I got from the surface rhythmic character of the passage, much as children respond to the concrete level character of the William Tell Overture. Part of what fascinates me about certain pieces of music is how they are able to generate a sense of goal-directedness when their surface attributes seem to be more "physical motion" than anything else. In particular, given my premise that beginning-accented rhythms connote non-teleological motion, and my observation that such rhythms do have that flavor in music, how does something like that prelude's middle section nevertheless project such a strong sense of goal-directedness and teleology? The answer toward which I have been groping is that what determines the section's overall character is the rhythmic groupings not on the surface level -- which produce one's initial sense of what is happening -- but several levels of rhythmic grouping up from the base level. (And what is fascinating about each piece is how the composer structures the rhythmic groupings at each level of structure of the piece, how he marshals the phrase lengths, the note durations, the pitch sequences, etc., in order to create hierarchical levels of rhythm that add up to a giant end-accented pattern that is the dynamic basis for the piece's goal-directed pull on the listener.)

I have also noticed that melodic factors -- repetition, sequence, ascent or descent -- play a big role in how rhythmic groupings relate to one another as stressed or unstressed. For instance, in the standard "My Heart Stood Still," the melody of the last 8 measures descends down to the tonic during the last 4 measures, which definitely imparts something like the denouement to the climax in a novel. Melodically and rhythmically, the song ends with an anti-climax, an unstressed phrase. Yet, in Frank Sinatra's classic recording (see "The Concert Sinatra"), the last 4 measures take an upward turn to the tonic an octave higher than the original version of the song. By "re-writing" the song in this way, Sinatra has changed the import of both the melody and the rhythm. Rather than subsiding into calm, contented "acceptance" of one's loving feelings, the song projects an exuberant, passionate "assertion" of those feelings. Rhythmically and melodically, the upward turn in the melody transforms it into a heroic, over-the-top ending -- what I like to call "leaping off the cliff." Or "ascending into heaven." In the version I perform of "When I Fall in Love," I do something similar with the ending. I guess the purpose is to "sweep the girl off her feet." :-)

More later...

REB

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One reason I am going to spend so much time data-basing and analyzing songs and melodies is in order to be inductive in my generalizations and conclusions. Sure, as Michael notes, my approach will be "statistical" rather than "epistemological," and it doesn't really provide a normative principle for evaluating music -- more like a set of norms of practice. In a way, it's like how Aristotle approach ethics by observing what virtuous men did. He collected a lot of data and generalized from that -- though he was also carrying around with him a few pre-fab criteria of what qualified as virtuous. (How else could he sort out the virtuous men from the ones who weren't?)

Some of my pre-fab ideas are basically just categories, conceptual file folders into which to toss the myriad melodies I'm going to examine. For instance, there are ABA and AABA song forms, a no-brainer -- sort and toss, sort and toss. There are songs that have primarily upward trending melodies vs. songs of the opposite kind -- and some of the most interesting songs that have sections with tendencies counter to the other sections, as well as songs with what Meyer calls "Sisyphean" sequences, where a phrase moves upward and back down partway, then up higher and back down partway, so that the overall trend is upward melodically, but with a sense of a zig-zag process of one step backward (or downward) for every two steps forward (or upward). This definitely has "dramatic" implications, though of a strictly musical kind (in the absence of a congruent text to give it an extra-musical reference). I'll be using such file folders for sorting and analyzing melodies, and I expect to find out a lot just as an observer and cataloguer of data.

Some of my pre-fab ideas, though, are more like hypotheses relating musical attributes to emotional or philosophical meaning: upward trending melodies connote assertion as against acceptance, end-accented rhythmic groupings connote goal-directedness as against more mechanical or physical motion, etc. Another pre-fab idea I'm using is the idea of hierarchical structure and the need to analyze a piece on the different levels of its structure, in order to thoroughly understand how it works, how the different levels work together or against one another, etc. -- and to do this not just with regard to rhythm but also melody and harmony. These are ideas that seem intuitively obvious to me, but I don't accept them as truths without a LOT of verification, and not just from cherry-picked examples. So, rest assured that I will leave no turned stone unexamined!

At some point, I anticipate that the inductive and hypothetico-deductive approaches will intersect, and I'll get some sort of resolution to my speculations -- and in the process learn a great deal about songs and musical themes, both about what they mean, and how they mean it. In the meantime, I am all ears about Michael's "epistemology of music" approach. I hope he will share it with us, even in rough form. It may touch off a whole new avenue for approaching my own project.

all for now,

reb

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Roger,

Have no fear about what you are doing. After I get some time to organize my thoughts on musical epistemology, I will gladly share them with you in order for you to have much food for thought.

Your work on the statistical angle is greatly needed and my only intention is to help highlight and separate the issues.

Rock on and keep an eye out for me for when I catch my breath. I guarantee that you will not be disappointed.

Michael

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  • 5 years later...
Folks, some of the fall-out from the Jan. 14 LAON meeting is that a comment by Nathaniel Branden triggered my resolve to begin work on a book on music aesthetics. Or, some mixture of the music theory, music psychology, and music aesthetics.

The book is based on a lecture I gave in San Francisco in March 2004, and its title is "Serious Schmaltz and Passionate Pop: are there objective indicators of emotion in music?" (That's the subtitle from my talk, but the subtitle for the book is a carefully guarded trade secret. Sorry. :-) [....]

REB

Since my first post about this project in January 2006, I've delivered related talks several times -- twice in Nashville, Tennessee in 2008 and 2009 and in Las Vegas, Nevada in 2009 at the Free Minds summer seminar. My talk in Las Vegas was entitled "Toward an Objectivist Aesthetics of Music," and it contains a great deal of material that will appear in the book, as well as in any future lecture course I will market on CDs and DVDs. I am planning to deliver it again locally in Middle Tennessee, and to apply some of the ideas pedagogically to music students at local colleges and universities. (I already passed along some of those ideas last fall to students at Middle Tennessee State University.)

Here are the four opening paragraphs of my 2009 talk:

It’s generally accepted that the gold standard in aesthetics is: can you explain emotion and meaning in music? Or as Rand herself phrased it: “Why does music make us experience emotions?” And it’s often suggested that Rand’s aesthetic theory fails on just that account. I couldn’t disagree more.

Today I’m going to argue that Ayn Rand’s general insights about art help us go a long way toward answering the question of the emotional meaning of music. Most of the essentials are already present in her writings to apply directly to explaining the meaning and emotional power of music. I will show how a number of general statements she makes about art apply fully well to music, and that she thus did not over-generalize her aesthetics of literature, as some claim.

I am in essential agreement with Rand’s key aesthetic ideas, including virtually all of her general comments about literature. I see them as being pregnant with implications for music, and I build on them in my approach to music. So, I’ll start by borrowing from Rand’s definition of “art” and define “music” as: the selective re-creation of reality according to the composer’s metaphysical value-judgments and by means of the sounds produced by the periodic vibrations of a sonorous body. Or, more briefly, music is the form of art created by the use of sounds of a definite pitch and character.

I have absolutely no doubt that music is one of the arts subsumed by Rand’s definition of art as “re-creation of reality,” and that her definition of “art” is valid. But what intrigues me are several key statements in The Romantic Manifesto about art in general. Let me read each of them, and then pose the question of how each of them pertains to music.

I also think it might be of interest to OL readers to see this brief outline of my 2009 talk, and I'll quote my concluding paragraphs at the end:

I. Introduction -- Some Questions about Music as a Form of Art

A. How does music enable us to grasp metaphysical value-judgments directly, on the perceptual level?

B. How is it possible that anything in music can be "life as I see it"?

C. With what in music are you identifying -- and how can it be applied to your life?

II. Music as a Dynamic Art

A. The attributes of art and music.

B. The basis of our experience of entities, attributes, and actions in music.

III. How are Metaphysical Value-Judgments Embodied in Music?

A. Basic forms of embodied abstractions in music--e.g., Intelligible Universe, Achievability of Values and Happiness, Pursing Values/Striving, Pursuing Values vs. Being Driven, and their relation to upward and downward melody, major and minor harmony, and beginning-accented vs. end-accented rhythm.

B. Complexities of embodied abstractions in music--e.g., mixing kinds of melody, harmony, and rhythm to get poignant or conflicted effects (simultaeous contrast between layers, sequential contrast between sections, climaxes or peaks of melodic motion, "Sisyphean sequences" of melodic motion, etc.).

C. The Heroic or "benevolent" sense of life and the Byronic or "malevolent" sense of life in music--did Beethoven have a "malevolent universe" perspective? If so, in what respect, and what about Chopin and Rachmaninoff, for instance?

IV. Conclusion -- Future Developments in Objectivist Music Aesthetics

In conclusion, there is a lot more research and theorizing to do, and a lot of application of the results to the creation, performance, and consumption of music. But I am convinced that the Objectivist aesthetics can lead us to new insights and enhanced enjoyment in both the theory and practice of music, along the lines I have sketched here today. In regard to theory, I think that the most fertile direction for development is in research. If someone wants to ask what kind of research I envision, I’d be happy to comment. As for applied music aesthetics, it should

• help listeners better understand why their feelings are being stirred by music

• help performers optimize the expressiveness of their playing and singing

• help composers know how better to tug at the heartstrings of their listeners

• help critics to more insightfully evaluate the artistry by which composers and songwriters present emotional meaning in music

• and help teachers to better help listeners, performers, and composers approach their musical experience with more success and enjoyment.

I think that the ideas I’ve presented here today give us reason for optimism about the future of the theory and practice of music.

I used numerous diagrams and audio examples with my talk, so it obviously is not going to be an easy thing to put into book form, certainly not with an accompanying CD using samples for which I'd have to pay royalties. Some of the examples are probably available on YouTube for free listening. This is why the more likely way of disseminating this material is going to be through college lectures. Until I work out the royalties issue. Before then, there is also a lot of listening, analyzing, and writing to do--so, onward!

REB

P.S. -- It's hard to believe that it has been over FIVE YEARS since I announced the beginning of this project. Time flies like an arrow -- and fruit flies like a banana. :-)

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  • 1 year later...

For those who are interested in the latest slate of my writing/publishing projects, the book on music has slipped back to about 5th place. Here is what I'm currently working on:

1. The Logic of Liberty (essays on ethics and politics, most written before 1985, but including my new essay)

2. What They Didn't Teach Me in Music School (a career guide for musicians)

3. You Should Know Better--and You Can! (a guide to logic texts for college students)

4. Will the Real Apollo Please Stand Up? and other essays (all using tetrachotomy analysis)

5. Serious Schmaltz and Passionate Pop (music aesthetics and analysis of classical and popular music)

I'm planning to self-publish #1 this summer; it's done, and I'm just waiting for the JARS 6-month "embargo" to lapse. I'm hoping to finish #2 this spring and put it out in the summer, too. I am about 25 pages into #3, and it's a doozy, with lots of stuff critiquing Copi, Hurley, and Kelley on existential import, standard propositional form, and the laws of thought themselves. I still have a lot to write for #4, but it is about 2/3 done. I may bump it ahead of #3. And last but not quite (as the 5 year old son of a music colleague once said), I don't expect to have the music aesthetics and analysis book done for 3-5 years. I may publish the portion of it on Beethoven's music and sense of life a good bit sooner.

REB

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Will your new book deal with some of the neurological/physiological aspects of what makes music pleasant to listen to. In particular the neurological/physiological bases of the scales and chord structures. I have just about reached the point where I accept the octave (the 2:1 ratio of frequencies) as a genuine built in neurological property of human hearing. I would like to learn much more about this and you are just the man to make that possible.

Ba'al CHatzaf

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Will your new book deal with some of the neurological/physiological aspects of what makes music pleasant to listen to. In particular the neurological/physiological bases of the scales and chord structures. I have just about reached the point where I accept the octave (the 2:1 ratio of frequencies) as a genuine built in neurological property of human hearing. I would like to learn much more about this and you are just the man to make that possible.

Ba'al CHatzaf

BC, I think this has already been done about 150 years ago by the great German scientist Helmholtz. Check out his book "On the Sensations of Tone." It's available from Dover.

However, I am doing some pioneering work in the neurological/physiological aspects of what makes logicians accept the doctrine of Existential Import. Blockbuster stuff, I tell ya...stay tuned! :-)

REB

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Will your new book deal with some of the neurological/physiological aspects of what makes music pleasant to listen to. In particular the neurological/physiological bases of the scales and chord structures. I have just about reached the point where I accept the octave (the 2:1 ratio of frequencies) as a genuine built in neurological property of human hearing. I would like to learn much more about this and you are just the man to make that possible.

Ba'al CHatzaf

BC, I think this has already been done about 150 years ago by the great German scientist Helmholtz. Check out his book "On the Sensations of Tone." It's available from Dover.

However, I am doing some pioneering work in the neurological/physiological aspects of what makes logicians accept the doctrine of Existential Import. Blockbuster stuff, I tell ya...stay tuned! :-)

REB

All gold fish in my pocket are over 6 inches long. Does that mean there are gold fish in my pocket that are over 6 inches long?

Existential Import is the denial of the empty set. Mathematicians will not accept this because empty sets do exist. For example the set of all real numbers whose squares are negative.

Ba'al Chatzaf

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  • 2 months later...

BaalChatzaf, on 14 Jan 2013 - 17:17, said:

Roger Bissell, on 14 Jan 2013 - 17:07, said:

BaalChatzaf, on 14 Jan 2013 - 07:58, said:

Will your new book deal with some of the neurological/physiological aspects of what makes music pleasant to listen to. In particular the neurological/physiological bases of the scales and chord structures. I have just about reached the point where I accept the octave (the 2:1 ratio of frequencies) as a genuine built in neurological property of human hearing. I would like to learn much more about this and you are just the man to make that possible.

Ba'al CHatzaf

BC, I think this has already been done about 150 years ago by the great German scientist Helmholtz. Check out his book "On the Sensations of Tone." It's available from Dover.

However, I am doing some pioneering work in the neurological/physiological aspects of what makes logicians accept the doctrine of Existential Import. Blockbuster stuff, I tell ya...stay tuned! :-)

REB

All gold fish in my pocket are over 6 inches long. Does that mean there are gold fish in my pocket that are over 6 inches long?

Existential Import is the denial of the empty set. Mathematicians will not accept this because empty sets do exist. For example the set of all real numbers whose squares are negative.

Ba'al Chatzaf

1. As (incompletely) stated, "All gold fish in my pocket are over 6 inches long" is ambiguous. You can see this when you try to state its contradictory, which by Aristotelian logic should have an opposite truth value: "All gold fish in my pocket ARE NOT over 6 inches long." This is clearly ambiguous. We aren't told whether it is supposed to mean: "All gold fish in my pocket ARE NOT REAL THINGS that ARE OVER 6 INCHES" or: "All gold fish in my pocket ARE REAL THINGS that ARE NOT OVER 6 INCHES LONG." The first is clearly true; the subject term does not designate real things, and the predicate correctly notes that fact. The second is clearly false; the subject in fact does NOT designate real things, while the predicate incorrectly states that it does. Which means that when you try to unambiguously restate the original, you have two possibilities. When you contradict the first to read: "All gold fish in my pocket ARE REAL THINGS that are over 6 inches long," it is clearly false, as it should be as per Aristotle. And this is clearly different from what you get when you contradict the second to read: "All gold fish in my pocket ARE NOT REAL THINGS that are not over 6 inches long," which is clearly true, as again it should be. So, the issue of Existential Import is really a trumped-up non-issue born of the failure to specify what mode of existence -- independent reality vs. arbitrary mental construct -- is being predicated of the subject term. In normal conversation about the real world, we assume that real existence is being predicated. However, when speaking about things that do NOT exist in the real world, especially arbitrary mental constructs such as "the present King of France" (a la Bertrand Russell) or "all gold fish in my pocket" (a la Ba'al), we have to deliberately step back and make it clear that we are NOT predicating real existence...unless we actually intend to, of course. For instance, if we had accumulated reports from people about their arbitrary mental posits of what gold fish might be in my pocket, we could then coherently say: "All gold fish in my pocket are arbitrary mental constructs that are over 6 inches long" (which might or might not be true, just as: "All sea serpents are imaginary reptiles that are green and scaly" might or might not be true), or: "All gold fish in my pocket ARE NOT arbitrary mental constructs that are over 6 inches long (which again might or might not be true, since some of them might be arbitrary mental constructs that are NOT over 6 inches long). But whichever the case, it means that in cases where the subject is non-existent, the statement must be rewritten with additional words in order to make explicit what we are asserting about the subject. Then (and only then) can we avoid the conundrums that have fueled scores of confusing, inconclusive essays and unnecessary restrictions on the applicability of Aristotle's Square of Opposition and his rules of immediate inference.

2. Empty rooms and empty containers exist, but empty sets do NOT exist. A set is not a container. A set is a collection of things that results from the ~mental~ collecting of some specified things. If there are no things of a particular kind to collect, then there is no collecting, no collection, and ~no set~. Modern logic's embracing of the empty set is very similar to, and just as destructive philosophically as, the idea that the universe could be devoid of things that exist. The universe is not a place or a container that holds all of the things that exist. It ~just is~ the sum total of all those things. If nothing existed, there would be no sum total and no universe. (But even to say "if" in this case is a mistake. The whole notion that the universe's existence is "radically contingent," that it's metaphysically possible that nothing could have existed, is often argued in terms of the universe being a set of things that exist, and that that "set" could have been "empty." Because after all, we can have "empty sets," don't you know.)

REB

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1. As (incompletely) stated, "All gold fish in my pocket are over 6 inches long" is ambiguous. You can see this when you try to state its contradictory, which by Aristotelian logic should have an opposite truth value: "All gold fish in my pocket ARE NOT over 6 inches long." This is clearly ambiguous. We aren't told whether it is supposed to mean: "All gold fish in my pocket ARE NOT REAL THINGS that ARE OVER 6 INCHES" or: "All gold fish in my pocket ARE REAL THINGS that ARE NOT OVER 6 INCHES LONG." The first is clearly true; the subject term does not designate real things, and the predicate correctly notes that fact. The second is clearly false; the subject in fact does NOT designate real things, while the predicate incorrectly states that it does. Which means that when you try to unambiguously restate the original, you have two possibilities. When you contradict the first to read: "All gold fish in my pocket ARE REAL THINGS that are over 6 inches long," it is clearly false, as it should be as per Aristotle. And this is clearly different from what you get when you contradict the second to read: "All gold fish in my pocket ARE NOT REAL THINGS that are not over 6 inches long," which is clearly true, as again it should be. So, the issue of Existential Import is really a trumped-up non-issue born of the failure to specify what mode of existence -- independent reality vs. arbitrary mental construct -- is being predicated of the subject term. In normal conversation about the real world, we assume that real existence is being predicated. However, when speaking about things that do NOT exist in the real world, especially arbitrary mental constructs such as "the present King of France" (a la Bertrand Russell) or "all gold fish in my pocket" (a la Ba'al), we have to deliberately step back and make it clear that we are NOT predicating real existence...unless we actually intend to, of course. For instance, if we had accumulated reports from people about their arbitrary mental posits of what gold fish might be in my pocket, we could then coherently say: "All gold fish in my pocket are arbitrary mental constructs that are over 6 inches long" (which might or might not be true, just as: "All sea serpents are imaginary reptiles that are green and scaly" might or might not be true), or: "All gold fish in my pocket ARE NOT arbitrary mental constructs that are over 6 inches long (which again might or might not be true, since some of them might be arbitrary mental constructs that are NOT over 6 inches long). But whichever the case, it means that in cases where the subject is non-existent, the statement must be rewritten with additional words in order to make explicit what we are asserting about the subject. Then (and only then) can we avoid the conundrums that have fueled scores of confusing, inconclusive essays and unnecessary restrictions on the applicability of Aristotle's Square of Opposition and his rules of immediate inference.

2. Empty rooms and empty containers exist, but empty sets do NOT exist. A set is not a container. A set is a collection of things that results from the ~mental~ collecting of some specified things. If there are no things of a particular kind to collect, then there is no collecting, no collection, and ~no set~. Modern logic's embracing of the empty set is very similar to, and just as destructive philosophically as, the idea that the universe could be devoid of things that exist. The universe is not a place or a container that holds all of the things that exist. It ~just is~ the sum total of all those things. If nothing existed, there would be no sum total and no universe. (But even to say "if" in this case is a mistake. The whole notion that the universe's existence is "radically contingent," that it's metaphysically possible that nothing could have existed, is often argued in terms of the universe being a set of things that exist, and that that "set" could have been "empty." Because after all, we can have "empty sets," don't you know.)

REB

The set of four sided triangles is as real as the set of three sided triangles. Sets do not exist in the physical sense. They are spooks generated by electro-chemical processes taking place in our bodies.

Ba'al Chatzaf

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BaalChatzaf, on 25 Mar 2013 - 08:23, said:

Roger Bissell, on 24 Mar 2013 - 20:38, said:

BaalChatzaf, on 14 Jan 2013 - 17:17, said:

1. As (incompletely) stated, "All gold fish in my pocket are over 6 inches long" is ambiguous. You can see this when you try to state its contradictory, which by Aristotelian logic should have an opposite truth value: "All gold fish in my pocket ARE NOT over 6 inches long." This is clearly ambiguous. We aren't told whether it is supposed to mean: "All gold fish in my pocket ARE NOT REAL THINGS that ARE OVER 6 INCHES" or: "All gold fish in my pocket ARE REAL THINGS that ARE NOT OVER 6 INCHES LONG." The first is clearly true; the subject term does not designate real things, and the predicate correctly notes that fact. The second is clearly false; the subject in fact does NOT designate real things, while the predicate incorrectly states that it does. Which means that when you try to unambiguously restate the original, you have two possibilities. When you contradict the first to read: "All gold fish in my pocket ARE REAL THINGS that are over 6 inches long," it is clearly false, as it should be as per Aristotle. And this is clearly different from what you get when you contradict the second to read: "All gold fish in my pocket ARE NOT REAL THINGS that are not over 6 inches long," which is clearly true, as again it should be. So, the issue of Existential Import is really a trumped-up non-issue born of the failure to specify what mode of existence -- independent reality vs. arbitrary mental construct -- is being predicated of the subject term. In normal conversation about the real world, we assume that real existence is being predicated. However, when speaking about things that do NOT exist in the real world, especially arbitrary mental constructs such as "the present King of France" (a la Bertrand Russell) or "all gold fish in my pocket" (a la Ba'al), we have to deliberately step back and make it clear that we are NOT predicating real existence...unless we actually intend to, of course. For instance, if we had accumulated reports from people about their arbitrary mental posits of what gold fish might be in my pocket, we could then coherently say: "All gold fish in my pocket are arbitrary mental constructs that are over 6 inches long" (which might or might not be true, just as: "All sea serpents are imaginary reptiles that are green and scaly" might or might not be true), or: "All gold fish in my pocket ARE NOT arbitrary mental constructs that are over 6 inches long (which again might or might not be true, since some of them might be arbitrary mental constructs that are NOT over 6 inches long). But whichever the case, it means that in cases where the subject is non-existent, the statement must be rewritten with additional words in order to make explicit what we are asserting about the subject. Then (and only then) can we avoid the conundrums that have fueled scores of confusing, inconclusive essays and unnecessary restrictions on the applicability of Aristotle's Square of Opposition and his rules of immediate inference.

2. Empty rooms and empty containers exist, but empty sets do NOT exist. A set is not a container. A set is a collection of things that results from the ~mental~ collecting of some specified things. If there are no things of a particular kind to collect, then there is no collecting, no collection, and ~no set~. Modern logic's embracing of the empty set is very similar to, and just as destructive philosophically as, the idea that the universe could be devoid of things that exist. The universe is not a place or a container that holds all of the things that exist. It ~just is~ the sum total of all those things. If nothing existed, there would be no sum total and no universe. (But even to say "if" in this case is a mistake. The whole notion that the universe's existence is "radically contingent," that it's metaphysically possible that nothing could have existed, is often argued in terms of the universe being a set of things that exist, and that that "set" could have been "empty." Because after all, we can have "empty sets," don't you know.)

REB

The set of four sided triangles is as real as the set of three sided triangles. Sets do not exist in the physical sense. They are spooks generated by electro-chemical processes taking place in our bodies.

Ba'al Chatzaf

If a set is merely a non-physical entity (?) or mental collection of items, and the set is something (whatever-it-is) that is generated by physical brain processes, it still has a nature, and it still has contents, whether or not those contents correspond to physical reality. To carry out a process of mentally collecting items and forming a set, the brain has to operate on the products of other brain products, whether those generated from currently perceived items or those generated from items that have been remembered or whatever. Your brain can produce a mental image of a green triangle, and so it can also produce a set containing one or more such imaginary items. Green-ness and three-sidedness ~can~ coexist in reality, even if they in fact ~may not~. But even though your brain cannot produce a mental image of a four-sided triangle, because four-sidedness and three-sideness do not and cannot coexist in reality, and therefore such a thing is not imaginary but arbitrarily posited (and contradictory), nonetheless the set of such impossible items would not be empty, but full of arbitrary posits of that kind. As you say, the two sets are equally real (equally brain-generated, that is), but the latter is no more empty than the former. The difference is that the mentally constructed (brain generated) contents of the former may but need not exist in reality, while the mentally constructed (brain generated) contents of the latter cannot exist in reality. Neither of them is an ~empty~ set, however, unless we arbitrarily define "set" as that which contains ~real~ (mind-independent) items. But even then, we are ruling out empty sets, since something which does not contain real items is by definition ~not~ a set.

REB

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The empty set is as much something as a non-empty set. It is a collection which does not have any elements. Think of your chest of drawers after all the socks are in the washing machine. The set of socks in your chest of drawers is empty (no doubt a temporary state of affairs).

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BaalChatzaf, on 25 Mar 2013 - 15:54, said:

The empty set is as much something as a non-empty set. It is a collection which does not have any elements. Think of your chest of drawers after all the socks are in the washing machine. The set of socks in your chest of drawers is empty (no doubt a temporary state of affairs).

There is no set of socks in my empty chest of drawers. The set of socks is the groups of socks, wherever they are, whether they are in some container or not. The chest of drawers is no more an empty set of socks than the washing machine is a full set of socks. Those containers are just where the socks are or aren't. The socks themselves are the set, and wherever they aren't, there is no set of socks, full or empty.

A set is not a container that either has things in it or does not. A set ~is~ the collected totality of things mentally or physically gathered, whether or not it is also in a container of some kind.

A set is not like a bucket or a room or a silverware chest. A silverware chest with no knives, forks, or spoons in it is not an "empty silverware set," any more than it is an "empty set of square circles." The silverware set ~is~ the collected totality of knives, forks, or spoons, not the thing that they are placed in.

A concept, which is a set of like things, ~is~ those things ~as the mind/brain holds them to be a group of similars~. We speak of the "content" of concepts, but that does not mean that concepts are empty forms or containers into which we pour mental contents of one kind or another. Concepts ~are~ the things that are ~mentally formed into~ a single unit, the group(ed together) similar things. The form is the grouped content, not some prior existing container into which the grouped content is poured.

The same is true for sets, but only more generally, since sets are not necessarily a grouping of similars, just a grouping of things selected for attention on ~some~ basis or other.

REB

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A set is not a container that either has things in it or does not. A set ~is~ the collected totality of things mentally or physically gathered, whether or not it is also in a container of some kind.

A set is not like a bucket or a room or a silverware chest.

I disagree with the first sentence. I would agree with the second sentence if "or physically" were deleted. I agree with the third sentence.

A set is a "mental container" with membership criteria. It is analogous to a physical container with a label on it. It isn't physical but can be depicted as such, e.g. in Venn diagrams.

An empty set is a set with no members because nothing meets the criteria, e.g. buildings more than 100 miles tall. Membership criteria that are contradictory, e.g. a triangle with 4 sides, makes an empty set.

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As I recall my math and logic training, a set need not have membership criteria in any interesting sense. A teacher once used the example of the set containing the Eiffel Tower, her dog and the number five. No conceptual thread connects these three objects (except the trivial one of being one of these three objects), but they make up a set just the same.

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I'd like a clarification on whether we are talking about ~an~ empty set or ~the~ empty set....though I'm not sure what difference the answer would make. This all seems incredibly bizarre and nonsensical to me.

1. If my (designated) sock drawer is empty, then according to what I am reading in this thread, it should contain an infinite number of empty sets, not just the (temporarily) empty set of my socks which are presently in the clothes dryer. I.e., it should contain (?) the empty set of my socks, but also the empty set of my wife's socks, the empty set of green triangles, the empty set of round triangles, etc. ad infinitum.

Even worse, it seems to follow that an infinity of empty sets is ~everywhere~ that the things they don't contain (!) aren't! Not only in empty spaces, but even in spaces that are completely filled with other things!

Doesn't it seem cognitively bizarre to populate the world -- both its empty areas and its full areas -- with an infinity of nothings? What is the cognitive purpose or value of doing this?

2.Is "the empty set" a master set that contains all of the particular empty sets? If so, is ~it~ empty, as its name suggests? In which case, how can it contain particular empty sets? I.e., if all the specific empty sets are things that "the empty set" contains as subsets, how can it be empty? On the other hand, if "the empty set" is ~not~ an uber-set containing all of the particular empty sets, then what is it, what is its relation to the infinity of specific empty sets, and what possible use does it fulfill?

I don't think that any of you have a good justification for the empty set. (Though I would not say you have ~no~ good justification for the empty set. No good justification is not something that you can have. :-)

REB

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I'd like a clarification on whether we are talking about ~an~ empty set or ~the~ empty set....though I'm not sure what difference the answer would make. This all seems incredibly bizarre and nonsensical to me.

1. If my (designated) sock drawer is empty, then according to what I am reading in this thread, it should contain an infinite number of empty sets, not just the (temporarily) empty set of my socks which are presently in the clothes dryer. I.e., it should contain (?) the empty set of my socks, but also the empty set of my wife's socks, the empty set of green triangles, the empty set of round triangles, etc. ad infinitum.

Even worse, it seems to follow that an infinity of empty sets is ~everywhere~ that the things they don't contain (!) aren't! Not only in empty spaces, but even in spaces that are completely filled with other things!

Doesn't it seem cognitively bizarre to populate the world -- both its empty areas and its full areas -- with an infinity of nothings? What is the cognitive purpose or value of doing this?

2.Is "the empty set" a master set that contains all of the particular empty sets? If so, is ~it~ empty, as its name suggests? In which case, how can it contain particular empty sets? I.e., if all the specific empty sets are things that "the empty set" contains as subsets, how can it be empty? On the other hand, if "the empty set" is ~not~ an uber-set containing all of the particular empty sets, then what is it, what is its relation to the infinity of specific empty sets, and what possible use does it fulfill?

I don't think that any of you have a good justification for the empty set. (Though I would not say you have ~no~ good justification for the empty set. No good justification is not something that you can have. :-)

REB

There is only one empty set. Two sets are equal if and only if being an element of one implies being an element of the other and vica versa. This is the same as saying not being an element of one implies not being an element of the other and vica versa.

So "two" empty sets are the same hence there is only one.

x <-> y if and only if -x <-> -y

Ba'al Chatzaf

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I'd like a clarification on whether we are talking about ~an~ empty set or ~the~ empty set....though I'm not sure what difference the answer would make. This all seems incredibly bizarre and nonsensical to me.

Is there only one number 5 or are there lots of them? Same answer. 4 + 1 = 5. 7 - 2 = 5. How many 5's are there?

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OK, so the one and only "empty set" is everywhere. Not just the "empty set" of my socks in my presently empty sock drawer, not just the "empty set" of my wife's socks in my presently empty sock drawer, nor the "empty set" of blue triangles or round triangles in my presently empty sock drawer, but also all of those and infinitely more that are all ~that same, one and only~ set that is in my sock drawer when my socks ~are~ there! And everywhere else in the universe!

Nothing -- i.e., the "empty set" -- is everywhere!

Now, I do agree that everything (that exists) is somewhere in particular, and in that sense there is not any thing (that exists) that is everywhere in general. But that is radically different from saying that "nothing" in the sense of ~that which does NOT exist~ IS everywhere -- which is what set theory's "empty set" construct implies.

My socks can and are absent from nearly everywhere in the universe. But that does not mean that their "non-existence" ~is~ here, there, and (nearly) everywhere. It only means that ~they~ ~aren't~ in those places! And in general, each and every thing is only where it is, and not where it isn't.

So, unlike 5's, of which there is the abstraction and the many specific instances of 5 things, there isn't one "empty set," and there aren't ~many~ "empty sets." There is no nothing. There is only something that is. When we say "there is nothing in my sock drawer," we aren't saying "there is an empty set" in that drawer, or that the drawer "contains nothing." We are saying that it ~doesn't contain anything~, that anything which it might contain is not there, but somewhere else. There is a vast difference between these two perspectives. One of them leads to Reification of the Zero, and the other affirms the Primacy of Existence.

REB

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I'd like a clarification on whether we are talking about ~an~ empty set or ~the~ empty set....though I'm not sure what difference the answer would make. This all seems incredibly bizarre and nonsensical to me.

Is there only one number 5 or are there lots of them? Same answer. 4 + 1 = 5. 7 - 2 = 5. How many 5's are there?

The cardinal number 5 is the class of all sets that can be put into 1-1 correspondence with the set of marks {a, b, c, d, e}.

Please see: http://en.wikipedia.org/wiki/Cardinal_number

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Point 2 in #42 above is Russell's paradox: if every class is a member of itself (as Frege had laid down as an axiom), what do you do with the class of all classes having no members? Is it a member of itself? Your insight is admirable but a century late.

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Hi Roger,

In the significant technical sense for logic and mathematics, sets are defined implicitly in an assembly of axioms, such as these. This approach to set theory yields the desirable result that certain sets that one can say, such as “the set of all sets” or “the set of all cardinal numbers” do not exist. They are not licensed by the axioms. As we see from the ZF axioms, the null set is said to exist in the sense that sets exist. I would suggest considering whether you want to affirm the existence of any sets as implicitly defined by these axioms and as used in logic and mathematics today. Or consider whether you would aim for discerning a different notion of mathematical and logical existence that is some specific sort rarified in comparison to physical-object existence, then see if you want to deny that sort of existence for the null set, but not all others.

Axiomatic set theory is where set theory is in our era. These are sets in a sense more than collection, which latter is a legitimate common usage of the term set, but not what matters, for example, in the set-theoretic formalization of part of group theory or of measurement theory. I encourage you to find out what those symbols mean in those axiom statements and be able to read in plain English what those axioms say in their symbolic notation. They are plain English sentences (or plain German, . . .).

Stephen

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Stephen Boydstun, on 27 Mar 2013 - 09:41, said:

Hi Roger,

In the significant technical sense for logic and mathematics, sets are defined implicitly in an assembly of axioms, such as these. This approach to set theory yields the desirable result that certain sets that one can say, such as “the set of all sets” or “the set of all cardinal numbers” do not exist. They are not licensed by the axioms. As we see from the ZF axioms, the null set is said to exist in the sense that sets exist. I would suggest considering whether you want to affirm the existence of any sets as implicitly defined by these axioms and as used in logic and mathematics today. Or consider whether you would aim for discerning a different notion of mathematical and logical existence that is some specific sort rarified in comparison to physical-object existence, then see if you want to deny that sort of existence for the null set, but not all others.

Axiomatic set theory is where set theory is in our era. These are sets in a sense more than collection, which latter is a legitimate common usage of the term set, but not what matters, for example, in the set-theoretic axiomatization of group theory or measurement theory. I encourage you to find out what those symbols mean in those axiom statements and be able to read in plain English what those axioms say in their symbolic notation. They are plain English sentences (or plain German, . . .).

Stephen

Thanks, Stephen, for the clarification and the suggestions.

Actually, I ~have~ been working along another avenue. I've been considering number as relational, and I've been considering how the number zero functions in the context of each of the types of numbers (counting #'s, integers, real #'s). If I interpret your use of the term "rarified" correctly, I've been doing what you suggest -- seeing numbers as abstractions of relationships between real things.

I don't want to throw out the accurate but mislabeled/misconceptualized results of modern math, any more than I want to throw out the accurate but mislabeled/misconceptualized results of modern physics. I just want to understand them in the same manner that I understand abstractions more generally, as per Rand's theory of concepts -- and place them in my knowledge hierarchy accordingly.

I've been listening to (ARI lecturer) Pat Corvini's lectures on number (total of 6, over 2 years), and she has a lot of insights, though she does not discuss zero specifically. She does, however, address the issue of the supposed one-to-one correspondendence between infinite "sets," and she pretty well demolishes the modern notion that there is, for instance, the same number of counting numbers as even numbers. (We've discussed that here previously, to no avail.)

Thanks again.

REB

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